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This paper is a contribution to the study of Borel equivalence relations on standard Borel spaces (i.e., Polish spaces equipped with their Borel structure). In mathematics one often deals with problems of classification of objects up to some notion of equivalence by invariants. Frequently these objects can be viewed as elements of a standard Borel space X… (More)

- Author R. Dougherty, Steve Jackson, Alexander S. Kechris, Randy Dougherty
- 2008

We study the structure of the equivalence relations induced by the orbits of a single Borel automorphism on a standard Borel space. We show that any two such equivalence relations which are not smooth, i.e., do not admit Borel selectors, are Borel embeddable into each other. (This utilizes among other things work of Effros and Weiss.) Using this and also… (More)

(A) A Polish metric space is a complete separable metric space (X, d). Our first goal in this paper is to determine the exact complexity of the classification problem of Polish metric spaces up to isometry. Our work was motivated by a recent paper of Vershik [1998], where the author remarks (in the beginning of Section 2): “The classification of Polish… (More)

- Greg Hjorth, Alexander S. Kechris
- Ann. Pure Appl. Logic
- 1996

The study of dynamical systems has its origins in the classical mechanics of Newton and his successors. That theory concerns the behavior of solutions of certain differential equations on manifolds. The modern theory has flourished in many directions that are best described by focusing on different features of the classical systems. For example, keeping… (More)

We study topological properties of conjugacy classes in Polish groups, with emphasis on automorphism groups of homogeneous countable structures. We first consider the existence of dense conjugacy classes (the topological Rokhlin property). We then characterize when an automorphism group admits a comeager conjugacy class (answering a question of Truss) and… (More)

(A) We study in this paper some connections between the Fräıssé theory of amalgamation classes and ultrahomogeneous structures, Ramsey theory, and topological dynamics of automorphism groups of countable structures. A prime concern of topological dynamics is the study of continuous actions of (Hausdorff) topological groups G on (Hausdorff) compact spaces X.… (More)

- Alexander S. Kechris
- Bulletin of Symbolic Logic
- 1999

§1. I will start with a quick definition of descriptive set theory: It is the study of the structure of definable sets and functions in separable completely metrizable spaces. Such spaces are usually called Polish spaces. Typical examples are Rn, Cn, (separable) Hilbert space and more generally all separable Banach spaces, the Cantor space 2N, the Baire… (More)

This paper is a contribution to the study of Borel equivalence relations in standard Borel spaces, i.e., Polish spaces equipped with their Borel structure. A class of such equivalence relations which has received particular attention is the class of hyperfinite Borel equivalence relations. These can be defined as the increasing unions of sequences of Borel… (More)

- Greg Hjorth, Alexander S. Kechris
- J. Symb. Log.
- 1995

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. Association for… (More)